Betti numbers of modules of essentially monomial type

Chang, Shou-Te

doi:10.1090/S0002-9939-00-05235-7

Betti numbers of modules of essentially monomial type
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Proc. Amer. Math. Soc. 128 (2000), 1917-1926 Request permission

Abstract:

Let $R$ be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of $R$-modules of essentially monomial type. These modules are defined with respect to various $R$-regular sequences. For example, finite length modules of monomial type over regular local rings of dimension $n$ are modules of essentially monomial type with respect to $R$-regular sequences of length $n$. If a module is of essentially monomial type with respect to an $R$-regular sequence of length $n$, then the rank of its $i$-th syzygy is at least $\binom {n-1}{i-1}$ and its $i$-th Betti number is at least $\binom ni$.

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